The ratio of accelerations due to gravity g 1 : g 2 on the surfaces of two planets is 5: 2 and the ratio of their respective average densities ρ 1 : ρ 2 is 2: 1 . What is the ratio of respecti

Question

The ratio of accelerations due to gravity g 1 ​ : g 2 ​ on the surfaces of two planets is 5: 2 and the ratio of their respective average densities ρ 1 ​ : ρ 2 ​ is 2: 1 . What is the ratio of respective escape velocities v 1 ​ : v 2 ​ from the surface of the planets?

TestHub · Accepted Answer

Escape velocity from a planet. v 0 ​ = R 2 GM ​ ​ = R 2 2 GM ​ ​ R = 2 g R ​ According due to gravity g = R 2 GM ​ = R 2 G 3 4 ​ π R 3 ⋅ ρ ​ = 3 4 ​ G π R p ​ ∴ Radius R = 4 π Gρ 3 g ​ From Eqs. (i) and (ii), we get v e ​ = 2 g ⋅ 4 π Gρ 3 g ​ ​ = 2 3 ​ π Gρ g 2 ​ ​ Thus, v e ​ ∝ ρ ​ g ​ ∴ v r 2 ​ ​ v n ​ ​ = ρ 1 ​ ​ g 1 ​ ​ × g 2 ​ ρ 2 ​ ​ ​ = 2 5 ​ × 2 ​ 1 ​ (given, g 2 ​ g 1 ​ ​ = 2 5 ​ and ρ 2 ​ Q ​ = 2 : 1 ) = 2 2 ​ 5 ​

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